A conjecture by Church-Farb-Putman predicts that the rational cohomology of SL_n(Z) vanishes in high degrees. In this series of two talks, we will explain why one might believe that this conjecture is true and describe recent progress that has been made for SL_n(Z) and related groups. The first talk focuses on SL_n(Z). The second talk puts the results in a broader context of arithmetic groups.